\section{1.2. Differential operators on a variety} 
\begin{frame}[allowframebreaks]{1.2. }

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1.2. We drop the assumption that $X$ is affine. 

It follows from the last assertion in 1.1 that $X$ carries a unique sheaf of rings $\mathcal{D}_X$ whose restriction to every open affine $Y \subset X$ is $\mathcal{D}_Y$. 

By definition, this is the sheaf of germs of {\color{red}algebraic differential operators} on $X$, and its (regular) sections over $X$ are the algebraic differential operators on $X$. 

It is quasi-coherent over $\mathcal{O}_X$. 

Its restriction to $Y$ open in $X$ is $\mathcal{D}_Y$.

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{\color{red}Notebook.} 

1. What does a `not quasi-coherent' module over $\mathcal{O}_X$ look like? 

2. Does $D(Y)$ for affine open sets $Y$ in $X$ determine the sheaf $\mathcal{D}_X$? 

3. Are there `meromorphic sections' of $\mathcal{D}_X$ instead of `regular sections'?

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1K. 

A sheaf of $\mathcal{O}_X$-modules $F$ is {\color{red}quasi-coherent} if for every point $x \in X$, there exists an open neighborhood $U$ of $x$ such that $F|_U$ is isomorphic to the cokernel of a morphism of free $\mathcal{O}_U$-modules (possibly of infinite rank). 

If $X$ is affine, this means $F \cong \widetilde{M}$ for some $\mathcal{O}(X)$-module $M$.

A typical example of a sheaf that is {\color{red}not quasi-coherent} is the skyscraper sheaf. 

Let $X = \mathbb{A}^1_\mathbb{C}$, $x = 0$, and let $i: \{0\} \hookrightarrow X$ be the inclusion. 

Let $k(x) = \mathbb{C}$ be the residue field at $x$. 

The skyscraper sheaf $i_*\mathbb{C}$ is an $\mathcal{O}_X$-module via the natural map $\mathcal{O}_X \to i_*\mathbb{C}$. 

But it is not quasi-coherent because its stalk at $0$ is $\mathbb{C}$, while for any $\mathcal{O}_X(U)$-module $M$, the sections $M$ over $U$ cannot produce a stalk that is a 1-dimensional $\mathbb{C}$-vector space in a way compatible with localization unless $M = 0$. 

More precisely, if $i_*\mathbb{C}$ were quasi-coherent, it would be of the form $\widetilde{M}$ for some $\mathbb{C}[x]$-module $M$, but then $M_x = M_{(x)}$ would have to be $\mathbb{C}$, which is impossible since localization of a $\mathbb{C}[x]$-module at $x$ cannot kill all positive-degree elements unless $M$ is torsion and not finitely generated in a way that violates quasi-coherence.

Another example is the sheaf $\mathbb{Z}_X$ of constant integer-valued functions (with $\mathcal{O}_X$-module structure via $\mathcal{O}_X \to \mathbb{Z}_X$, which is not natural), but this is not even a module in the usual sense unless twisted. 

A better example is a sheaf supported on a non-open, non-closed set, or a sheaf with non-finitely generated stalks that do not arise from localization.

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2K. 

Yes, the rings $\mathcal{D}(Y)$ for affine open subsets $Y \subset X$ completely determine the sheaf $\mathcal{D}_X$. 

This is because $\mathcal{D}_X$ is a sheaf, and the affine open subsets form a basis for the Zariski topology on $X$. 

A sheaf on a topological space is uniquely determined by its values on a basis, provided the gluing conditions are satisfied.

In this case, for any open set $U \subset X$, we can write $U = \bigcup_i Y_i$ with $Y_i$ affine open. 

Then by the sheaf property,
\[
\mathcal{D}_X(U) = \varprojlim_{Y \subset U,\ Y\ \text{affine}} \mathcal{D}(Y),
\]
or more precisely, $\mathcal{D}_X(U)$ consists of families of elements $(s_i) \in \prod_i \mathcal{D}(Y_i)$ that agree on overlaps $Y_i \cap Y_j$, which can be covered by affine opens as well.

Moreover, the compatibility condition (1.1): $\mathcal{D}(Y) = \mathcal{O}(Y) \otimes_{\mathcal{O}(X)} \mathcal{D}(X)$ for $Y$ affine open in affine $X$, ensures that the transition maps are well-defined and satisfy the cocycle condition. 

Therefore, the data $\{ \mathcal{D}(Y) \mid Y \subset X\ \text{affine open} \}$ with restriction maps defines a unique quasi-coherent sheaf $\mathcal{D}_X$ on $X$.

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3K. 

Yes, there are `meromorphic' sections of $\mathcal{D}_X$, and they form a sheaf of rings known as the sheaf of {\color{red}meromorphic differential operators} or the {\color{red}total ring of fractions} of $\mathcal{D}_X$. 
Here's how this is defined:

Let $X$ be a smooth complex algebraic variety. 

Since $X$ is smooth and irreducible (as it is of pure dimension), the structure sheaf $\mathcal{O}_X$ has no zero divisors, and its stalk at the generic point $\eta$ is the function field $K(X)$.

The sheaf $\mathcal{D}_X$ is a filtered sheaf of rings with $\mathcal{D}_X^{(0)} = \mathcal{O}_X$, and each $\mathcal{D}_X^{(m)}$ is a coherent $\mathcal{O}_X$-module. 

Because $\mathcal{O}_X$ is an integral domain, $\mathcal{D}_X$ is also torsion-free over $\mathcal{O}_X$, and hence injects into its localization at the generic point.

We define the {\color{red}sheaf of meromorphic differential operators} on $X$, denoted $\mathcal{D}_{X}^{\mathrm{mer}}$, by:
\[
\mathcal{D}_{X}^{\mathrm{mer}}(U) = \mathcal{D}_X(U) \otimes_{\mathcal{O}_X(U)} K_X(U)
\]
for any open set $U \subseteq X$, where $K_X$ is the constant sheaf associated to the function field $K(X)$. 

Equivalently, $\mathcal{D}_{X}^{\mathrm{mer}} = \mathcal{D}_X \otimes_{\mathcal{O}_X} K_X$.

This is a subsheaf of the sheaf of endomorphisms of $K_X$ over $\mathbb{C}$, consisting of those $\mathbb{C}$-linear endomorphisms that are differential operators when restricted to regular functions (with poles allowed).

More precisely, a section of $\mathcal{D}_{X}^{\mathrm{mer}}$ over $U$ is a $K(X)$-linear combination of elements of $\mathcal{D}_X(U)$, or equivalently, a differential operator on $K(X)$ that can be written locally as $P/f$ where $P \in \mathcal{D}_X(V)$ and $f \in \mathcal{O}_X(V) \setminus \{0\}$ for some affine open $V \subseteq U$.

In local coordinates $(x_1, \ldots, x_n)$ on an open set $U$, elements of $\mathcal{D}_{X}^{\mathrm{mer}}(U)$ can be expressed as finite sums
\[
\sum_{\alpha} \phi_\alpha(x) \partial^\alpha,
\]
where $\phi_\alpha(x) \in K(X)$ are rational functions (meromorphic functions), and $\partial^\alpha = \partial_{x_1}^{\alpha_1} \cdots \partial_{x_n}^{\alpha_n}$.

Thus, just as meromorphic functions are rational sections of $\mathcal{O}_X$, meromorphic differential operators are rational sections of $\mathcal{D}_X$. 

They play an important role in the study of D-modules, especially in comparison with analytic or formal settings, and in the Riemann-Hilbert correspondence.

Note: While $\mathcal{D}_X$ is a coherent $\mathcal{O}_X$-module, $\mathcal{D}_{X}^{\mathrm{mer}}$ is {\color{red}not quasi-coherent} in the usual sense (it is a constant sheaf tensored with a filtered ring), but it is a well-defined sheaf of rings containing $\mathcal{D}_X$ as a subsheaf.


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